“Alduin's Bane” is the Thirteenth Main Story Quest in The Elder Scrolls V: Skyrim
You get “Alduin's Bane” after completing the quest “Elder Knowledge”.
Leave the depths of Blackreach using the nearby elevator across from the crystal container that held the Elder Scroll. This will bring the Dragonborn back to the surface. Fast travel to the Throat of the World. Head to the marked location and listen to what Paarthurnax has to say about your possession of the Elder Scroll. When he finishes talking you will be prompted to open your inventory and read the Elder Scroll.
When you first arrive in the past, there is nothing you can do but what the Nord Hakon fight the dragon. He is quickly joined by Gormlaith. The pair of them slay the dragon and then talk of the plan that they have. A third Nord is revealed shortly thereafter: Felldir. He reveals that he brought the Elder Scroll in case it was needed in the fight against Alduin. After their talk Alduin approaches and lands on the Mountain. Just after he talks the heroes use Dragonrend on him and the Dragonborn learns the words to use it themselves.
Now the Dragonborn must face the eldest Dragon. Equip and use Dragonrend to bring Alduin down on the ground. Immediately begin attacking him with the best equipment and spells on hand. When Alduin starts flying again, once again bring him down with Dragonrend. This is a brutal battle where nothing should be held back. Make use of all potions. Watch for Alduin to be taking off, hovering or swooping in to hit him with Dragonrend. Be sure to use the maximum power for Dragonrend in all cases.
While Alduin is flying, he will either use Firebreath or drop flaming rocks on you. When he is on the ground, he will loose Firebreath as well. This is in addition to snapping and swiping at you.
It will only end when either the Dragonborn delivers the death blow to Alduin or Alduin savages the Dragonborn. He will take to the sky after that. He cannot be slain here apparently. Time to learn more about this.